Since the internal circuitry of the LM399(A) shunts extra current around the Zener, there is no way to decrease the awful noise spec by increasing current-- so you may as well run it at the 1.0mA current to get the best long term stability.
That´s a good point, I did not recognize this.
I made 2 measurements with LM399#19 out of my ageing PCB.
One with a 6k8 pull up to 14.1V stabilized from a battery. (around 1 mA)
the other with 6K8 + 1K in parallel (around 8 mA)
All packaged into a cookie tin can and amplified *10000 with a 0.1 ... 10 Hz 4th order bandpass.
So the scaling on the oscilloscope is 1sec/div on time axis and 0.5uV/div in y-axis.
So there is no visible difference between 1 mA and 8 mA.
On the other side I do not understand from where do you get the figure of 10uVpp of noise.
Measured with which bandwidth?
With correct setup of 0.1 ... 10Hz bandwidth I usually measure values from 2.5uVpp to 4uVpp as average with std deviation of 0.2 .. 0.4. Only one LM399 is around 7uVpp.
With best regards
Andreas